Milnor number at infinity, topology and Newton boundary of a polynomial function
For a complex polynomial \(f:\mathbb{C}^n\to \mathbb{C}\) the set of values where \(f\) fails to be a \(C^{\infty}\)-locally trivial fibration (the bifurcation set) includes not only the critical values of \(f\) but may also contain values corresponding to `critical points at infinity' [for a reference see, e.g., \textit{A. H. Durfee}, Prog. Math. 162, 345-360 (1996; Zbl 0919.32021)]. At such an atypical value at infinity the fibre is different from the generic one. The paper studies a topological invariant of fibres, namely Euler characteristic. The authors prove the formulas (similar to those of Ha, Le, Siersma, Tibar): \[ \chi(\text{generic fibre})=1+(-1)^n(\mu(f)+\lambda(f)) \] \[ \chi(\text{fibre over }b)=\chi(\text{generic fibre})+(-1)^n (\mu^a_f(b)+\lambda_f(b)), \] where the \(\mu\)'s are sums of local Milnor numbers and the \(\lambda\)'s measure singularities at infinity. Their definition of \(\lambda\) relies on the generalized Milnor number of Parusiński (see also works of Parusiński-Pragacz), so is well computable -- which is an advantage to earlier works, and is well demonstrated in many examples in the paper. In case \(f\) has non-degenerate Newton principal part at infinity they relate the Euler characteristic of the generic fibre to \(\nu(f)\), the Newton number of \(f\). This, together with the formulas above, gives the main result of the paper \(\mu(f)+\lambda(f)=\nu(f)\), which generalizes Kushnirenko's theorem and also some results of Cassou-Nogues.
scientific article; zbMATH DE number 1463421